Problem 72 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ 3 x^{2}+x-x(5-x)-2 $$

Step-by-Step Solution

Answer

The standard form of the expression is \(4x^2 - 4x - 2\).

Step 1: Expand the Expression

First, distribute the multiplication within the expression \(x(5-x)\). This gives us \(-5x + x^2\). Now the entire expression becomes: \[3x^2 + x - (5x - x^2) - 2\].

Step 2: Simplify Inside the Parentheses

Remove the parentheses by considering the subtraction: \(3x^2 + x - 5x + x^2 - 2\).

Step 3: Combine Like Terms

Combine the \(x^2\) terms: \((3x^2 + x^2) = 4x^2\). Next, combine the \(x\) terms: \((x - 5x) = -4x\). Include the constant term: \(-2\). Thus, the full expression in standard form is: \[4x^2 - 4x - 2\].

Key Concepts

Polynomial Simplification

Polynomial simplification is all about making an expression more understandable and easier to work with by removing any unnecessary parts. Think of it like tidying up a room.
First, in any expression, begin by expanding any terms where necessary. This is like unpacking boxes before arranging items neatly. In our case, multiplying was needed between the variable and the binomial:

• Expand: Multiply out expressions such as \( x(5-x) \) to become \(-5x + x^2\).

Next, eliminate any redundant parentheses by carefully dealing with operations like subtraction or addition. This means adjusting terms appropriately as they're brought outside the parentheses.
By following these steps, you get a clearer view of the entire expression, setting the stage for combining similar components together.

Standard Form

Standard form in polynomials involves arranging terms by their degree from highest to lowest. This helps in organizing and understanding complex polynomial expressions. In our example, the expression has been simplified to:

• Original: \(3x^2 + x - 5x + x^2 - 2\)
• Simplified: \(4x^2 - 4x - 2\)

In writing in standard form:

• Start with the term with the highest power of the variable. Here, it's \(4x^2\).
• Follow with the next lower degree term. In this case, \(-4x\).
• Finally, place the constant, \(-2\), last.

Using this format ensures anyone reviewing your work will immediately see the entire structure of the polynomial, greatly aiding in future calculations and comparisons.

Combining Like Terms

Combining like terms means adding or subtracting terms within an expression that have the same variable raised to the same power. Think of this as grouping similar items together in a drawer for easy access.
In our exercise, the terms needed combining:

• Combine \(x^2\) terms: Summing \(3x^2\) and \(x^2\) produces \(4x^2\).
• Next, the \(x\) terms: Combine \(x\) and \(-5x\) to get \(-4x\).

Remember, constants (like \(-2\)) are individual and don’t combine with variable terms. Gathering like terms simplifies the expression and highlights key components, making it easier to solve or further manipulate as needed.